- Thread starter
- #1

$$

d_2(\mathbf{x},\mathbf{y}) = \sum_{n = 1}^{n}|x_i - y_i|.

$$

Prove that the ball $B(\mathbf{a},r)$ has the geometric appearance indicated:

In $(\mathbb{R}^2,d_2)$, a square with diagonals parallel to the axes.

Take a ball $B((x,y),r)$ for $r > 0$. Let $(a,b)\in B((x,y),r)$.

By the metric, we have $||x-a| + |y-b|| < r$

Now what?